Problem: Let $f(x, y, z) = y^2z$ and $g(t) = (t, t^2, t^3)$. $h(t) = f(g(t))$ $h'(2) = $
Formula The multivariable chain rule says that $\dfrac{dh}{dt} = \nabla f(g(t)) \cdot g'(t)$. The $g'(t)$ part is how much a change in $t$ will cause the input to $f$ to move, and the $\nabla f(g(t))$ part is how much $f$ will change in response to this update to its input. [What's the intuition behind the formula?] Applying the formula We want to find $h'(2) = \nabla f(g(2)) \cdot g'(2)$. $\begin{aligned} &g(2) = (2, 4, 8) \\ \\ &g'(2) = (1, 2(2), 3(2)^2) = (1, 4, 12) \\ \\ &\nabla f = (0, 2yz, y^2) \\ \\ &\nabla f(g(2)) = \nabla f(2, 4, 8) = (0, 64, 16) \end{aligned}$ Substituting: $\begin{aligned} h'(2) &= (0, 64, 16) \cdot (1, 4, 12) \\ \\ &= 256 + 192 \\ \\ &= 448 \end{aligned}$ Answer $h'(2) = 448$